In $PG(3,q)$, a $(q^2+1)$-cap is an ovoid. A $(q^2+1)$-cap is a set of $q^2+1$ points, no three of which are collinear. What is the dual of an ovoid?
I know how to get the dual of a statement involving points, lines and incidence between them by interchanging the words "point" and "line". So a $(q^2+1)$-cap would correspond to a set of $q^2+1$ lines no three of which intersect at a given point. Is this the dual of an ovoid?
In 3 dimensions the duals of points are the planes, and the duals of lines are the lines. So in this case the dual of an ovoid is a set of $q^2+1$ planes, no three of which have a common line. (As the ovoid is a generalization of quadrics; and in classical geometry the dual of a quadric is the set of the tangent planes of a quadric.)